Ring (mathematics)
A ring is an algebraic structure defined in the similar as in the whole numbers of addition and multiplication with respect to each other and bracketing are compatible. The ring theory is a branch of algebra that deals with the properties of rings.
- 4.1 homomorphism
- 4.2 isomorphism
- 4.3 Example
- 5.1 Lower and Upper Ring
- 5.2 Ideal
- 5.3 factor ring
- 5.4 Basic Ring
- 6.1 divider and a zero divisor
- 6.2 invertibility, unit
- 6.3 Associated Elements
- 6.4 irreducibility
- 6.5 prime element
Naming
The name ring does not refer to something vividly Annular but to a structured group of elements into a whole. This sense is otherwise largely lost in the German language. Some older club names ( such as German Ring, White Ring, machinery ring ) or phrases such as " crime ring " have yet point to this meaning. The concept of the ring goes back to Richard Dedekind; the name ring, however, was introduced by David Hilbert. In special situations next to the name ring the designation area is common. Thus one finds in the literature rather the term integral domain integral domain instead.
Remark on the ring - Definition
Depending on the sub-region and textbook ( and to some extent depending on the chapter ) is understood to be a ring something different. Also slightly different then the definitions of morphisms as well as lower and upper structures. In mathematical terms, it is at these different ring terms to different categories.
Definitions
Ring
A ring is a set with two binary operations and so
- An abelian group,
- A semigroup is
- The distributive
The neutral element is called zero element of the ring.
A ring is called commutative if it is commutative with respect to multiplication, otherwise it is called a non-commutative ring.
Ring with unit ( unitary ring )
Has the semigroup in addition a neutral element, ie it is a monoid, then it is called a ring with one or unitary ring. Some authors consider to be a ring basically a ring with unity.
Commutative ring with one
In the commutative algebra rings are defined as commutative rings with unity.
Examples
- The integers and polynomial rings over a field are commutative rings with unity.
- The zero ring consisting of only one element, is also a commutative ring with unity ().
- The ring of even integers is a commutative ring without identity.
- The matrix ring is a non- commutative ring with unit ( the unit matrix).
Homomorphism
Ring homomorphism
For two rings and is called a mapping
Homomorphism ( homomorphism short ), if for all:
The core of a ring homomorphism is a two-sided ideal in.
A morphism of rings with unity must also meet even the condition that the identity element is mapped to the identity element:
Isomorphism
An isomorphism is a bijective homomorphism. The rings and are called isomorphic if there is an isomorphism from to. The rings have the same structure.
Example
The figure
Is a homomorphism of rings, but no homomorphism of rings with unity.
Lower and upper structures
Lower and upper ring
A subset of a ring is called a subring of when together with the two on restricted links of another ring. is exactly then a subring of when a subgroup under addition and is closed under multiplication, i.e.,
Even if a ring with unity, so that one does not necessarily have to be included. can also be a ring without identity - about - or have a different one. For rings with identity is requested by a subring that it also contains the identity element ( for this it is necessary but not always sufficient that the subring one based on this multiplicative neutral element contains ).
The average of sub-rings is another sub- ring and the lower ring of generated is defined as the average of all sub- comprehensive rings.
A ring is called top ring or extension of a ring when a subring of is.
Ideal
At a ring is called a subset of left ideal (or right ideal ) if:
- Is a group of.
- For all and is also (or ).
If both left-and right ideal, so called two-sided ideal or simply ideal.
Contains in a ring with one one ( left, right ) ideal of the One, so it covers the whole. Since an ideal is, is the only ( left, right ) ideal which contains the One. and are the so-called trivial ideals.
Factor ring
Is an ideal in a ring, then you can the set of cosets
Form. The link can always continue on for their commutativity; the link, but only if a two-sided ideal in is. If this is the case, then a ring with the induced linkages. He is called factor ring - spoken: modulo.
The ring homomorphism
Of an element assigns its coset has the core.
Base ring
In a ring with one of the subring generated by is called the base ring. From the thickness of the base ring results in the characteristic of the ring.
Special elements in a ring
Divider and a zero divisor
From two elements is called left divider ( divider links ) from, if one exists with. Then just as well in multiples of. Accordingly, one defines right divider ( divider right ) and left several times.
Commutative rings in a left divider is also a right-hand and vice versa. We write here also, if a divisor of is.
All elements of are ( right or left ) divisor of zero. The notion of ( right or left ) zero divider has a different definition. If this counts as a zero divisor, the sentence is true: An element is then precisely ( right or left ) divisor of zero if it is not shortened (right or left ).
Invertibility, unit
Exists in a ring with one-to- one element is an element such that ( respectively ) is considered, it is called a left inverse (respectively right inverse ) of. Has both left-and right inverse, it is called invertible or unit of the ring. The quantity of units of a ring with one commonly referred to, with or. forms with respect to the ring multiplication is a group - the unit group of the ring. If so is a skew field, is also commutative, so is a body.
In commutative rings with identity ( in particular, integrity rings ) is defined alternatively the units as those elements that share the one. The fact that one shares, ie namely that there are with.
Associated elements
Two elements are accurate and then associates the right if there is a single legal entity so. Links associated with a left unit.
If in a commutative ring with unity are the elements in the relationship and, then, and associated with each other. The laterality ( left, right) can therefore be omitted.
Associated awareness is an equivalence relation.
Irreducibility
An element that is neither left nor right unit unit is, irreducible, if there are no non- unit links and no non-law unit, so if it follows from the equation that links unit or legal entity is.
In commutative ring, it suffices to require that from always or follows.
Prime element
If neither left nor right unit unit, then called prime (or prime element ) if for all with and it follows that there are with.
In commutative ring, it is sufficient to demand: Is a non- unit equal to 0, then that means prime (or prime element ) if the following applies: It follows from or (also see main article: prime element ).
In a zero-divisor- free ring, every prime element is irreducible.
Other examples
The rational numbers with the usual addition and multiplication form a ring. There is in this case, not only, but also an abelian group, there is even a body; it involves the quotient field of the ring integrity.
No ring is the set of natural numbers, with the usual addition and multiplication, since the addition to the natural numbers is not invertible.
Other important examples of rings are residue class rings, polynomial rings and square matrices with a fixed dimension. In particular, residue class rings and square n × n matrices with n> 1 provide examples of rings which are not zero divisors.